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# Business Calculus MATH 135

Cal State Fullerton

GPA 3.92

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This 17 page Class Notes was uploaded by Gunner Price III on Wednesday September 30, 2015. The Class Notes belongs to MATH 135 at California State University - Fullerton taught by Scott Annin in Fall. Since its upload, it has received 27 views. For similar materials see /class/217025/math-135-california-state-university-fullerton in Mathematics (M) at California State University - Fullerton.

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Date Created: 09/30/15

Math 135 Midterm II Information Spring 2009 WHEN Monday March 23 RULES No books or notes You may bring a scienti c nongraphing calculator to use for the exam It is recommended that you ensure that your calculator has suf cient battery power because if your calculator fails during the exam you will have to continue the exam without it Please bring your own calculatorllll Do not show up expecting me to supply one to you If you don7t bring one you will have to take the exam Without a calculator REVIEW SESSIONS Saturday March 21 12 2 pm MH 285 Sunday March 22 3 5 pm MH 285 EXTRA OFFICE HOURS Thursday March 19 3 6 pm Friday March 20 11 2 pm Saturday March 21 2 3 pm Sunday March 22 1 3 pm Monday March 23 9 10 am and 11 1 pm COVERAGE The midterm will cover Chapter 2 Sections 1 6 and Chapter 3 Sections 1 2 STUDYING Here is an overview of the topics we have covered You should be comfortable with all of the following phrases below 0 Increasing function 0 Decreasing function 0 Critical value 0 Relative maximum minimum 0 First Derivative test 0 Concave up and Concave down ln ection points Second Derivative test Asympototes vertical and horizontal Intercepts z intercepts and y intercept Absolute maximum minimum Extreme Value Theorem 0 Maximum Minimum Problems areas perimeters volumes costs revenue pro t etc Marginal cost Marginal revenue Marginal pro t Di erentials dx Ax and dy f xd Exponential functions De nition of e Derivatives involving f em Logarithmic functions Properties of logarithms Derivatives involving f lnx It is not enough to simply know what all of the words and phrases above mean You need to be able to solve problems that involve the concepts that occur in the list above To help you l7ve made a list of some of the most important things that you should be able to do con dently without notes books etc by the time you reach the midterm THINGS TO BE ABLE TO DO Be able to nd the relative extrema of a function This includes nding the critical points and classifying them either with the First Derivative Test or the Second Derivative Test Given a graph of fx be able to sketch a rough graph of f Given a graph of f be able to sketch a rough graph of See Problems 85 90 on p 213 Be able to use the second derivative to nd in ection points and intervals of concave up and down for the graph of a function Be able to draw graphs for functions f with various characteristics on f f and f See Problem 1 in Group Work 4 Be able to nd vertical and horizontal asymptotes if any for a function Be able to nd z intercepts and y intercepts if any for a function Be able to sketch the graph of a function f and incorporate information about intercepts asymptotes intervals of increasedecrease relative extrema in ection points intervals of concave updown etc Be able to nd absolute maximum and absolute minimum values for a function fx on a speci ed domain Be able to solve optimization word problems involving topics such as area volume perimeter cost revenue pro t etc Be sure to review examples and problems in Section 25 carefully Be able to compute the marginal cost revenue or pro t if the corresponding cost function revenue function or pro t function are given Be able to use the marginal functions to approximate costs revenues and prof its Know how to compute di ferentials and be able to use them to approximate the values of a given function f at a given point Be able to sketch basic graphs of exponential functions Be able to take derivatives of functions involving fx cm This may re quire use of power product quotient sumdifference andor chain rules from Chapter 1 Be able to sketch basic graphs of logarithmic functions Be able to take derivatives of functions involving fz ln z This may re quire use of power7 product7 quotient7 sumdifference7 andor chain rules from Chapter 1 Know the properties of logarithms listed in Theorem 4 on p 323 and be able to apply them SOME ADVICE Review the group work especially Group Works 4567 and 7 The problems in the group work were written by the same person who will be writing your midterm7 and were deemed important enough to spend class time on Try re doing the group works blank copies are available on the webpage7 and then go back and check the solutions Do not simply read over my solutions Focus on the areas in which you are least comfortable If there is a particular section or two of the chapter that you are not as con dent on7 perhaps spend a little extra time thereimaybe try to re do a few extra problems from that section Find at least one other person in the class to study with When you are talking about mathematics7 you remember it better When you ask someone else a question7 it helps that person check their own understanding and also helps you to get your questions answered So don7t be afraid to ask questions Here7s an idea Find a study buddy and write them a quiz of some of the material Work out the answers to your quiz rst7 and then give the quiz to your buddy Your buddy can do the same for you Then exchange answers and see how well you each did Feel free to ask me as many questions as you want I would also be happy to pop quiz77 you on a topic For example7 you can come ask me to give you some functions to graph7 and you can practice right on the spot Or I can give you some word problems Whatever you want Basically7 l7m here to help and I want everybody to do well7 so please don7t be shy Practice Practice Practice 0 Take the sample midterm as if it was a real test no notes no books no help from friends no outside distractions PRACTICE Here is a list of some good problems from the book that you might like to practice with After that you can try the sample midterm that I posted Again print out the blank test and try itisolutions will be posted later but you should not read those until you have tried to solve the problems on your own Please do not expect the actual midterm to be the same as the sample midterm While some of the problems might be similar I make no promises in this regard and you should study everything well not just the stu that appears in the sample midterm Good Problems from the Book 21 91927697589 229 11 4753 243 5713152129475153 255 17 29516397 27 9192743 281 73943 31 71729394547597787 332 17 273745556983 H O 7575777775757575 Cf Math 135 Midterm I Information Spring 2009 WHEN Monday February 23 RULES No books or notes You may bring a scienti c nongraphing calculator to use for the exam It is recommended that you ensure that your calculator has suf cient battery power because if your calculator fails during the exam you will have to continue the exam without it REVIEW SESSIONS Saturday February 21 3 5 pm MH 285 Sunday February 22 12 2 pm MH 285 EXTRA OFFICE HOURS Thursday February 19 3 6 pm Friday February 20 II noon and 1 2 pm Saturday February 21 2 3 pm Sunday February 22 2 4 pm Monday February 23 9 10 am and 11 1 pm COVERAGE The midterm will cover Chapter 1 STUDYING Here is an overview of the topics we have covered You should be comfortable with all of the following phrases below 0 Slope intercept and point slope equations for a line 0 Function numerical graphical algebraic words 0 Left hand limit Right hand limit Two sided limit 0 Algebraic limits factoring rationalizing roots etc 0 Continuous functions at a point and on an interval 0 Average rate of change 0 Difference quotient It is not enough to simply know what all of the words and phrases above mean You need to be able to solve problems that involve the concepts that occur in the list above To help you l7ve made a list of some of the most important things that you should be able to do con dently without notes books etc by the time you reach Limit of a difference quotient lnstantaneous rate of change eg slope velocity etc Tangent line to a curve De nition of the derivative Derivative notations Prime notation and Leibniz notation Power rule Derivative rules for additionsubtraction Product rule Quotient rule Extended power rule Composition of functions Chain rule Higher derivatives application velocity and acceleration the midterm THINGS TO BE ABLE TO DO Be comfortable with rules of algebra particular with respect to taking roots laws of exponents factoringcancelling and general simplifying Be comfortable with the slope intercept and point slope equations for a line Be able to draw graphs of basic functions such as parabolas lines simple rational functions root functions etc Be able to write down the domain of a function f Be able to compute left hand right hand and two sided limits for a function f at a given point a by a using a graph b plugging in values of x close to a c doing algebraic manipulation of a function In particular you should be very comfortable with piece wise de ned functions which arise a lot Be able to decide if a limit exists and the value of the limit when it does You should be able to justify your answers with words and calculations as necessary Be able to decide if a given function f is continuous at a given point 1 Know that polynomials and rational functions are continuous at all points in their domains Be able to write down the difference quotient for a function and simplify it Be able to compute the derivative f of a given function f Also be comfort able with di erent notations for the derivative y or f or 5 or Be able to nd an equation of the tangent line to the graph of y f at a given point z 1 Know the Power Rule Product Rule Quotient Rule Extended Power Rule and Chain Rule for di ferentiation You should be able to apply these rules ef ciently to nd derivatives Be able to compute higher derivatives The main application is to position velocity and acceleration Be able to solve word problems about rates of change by using the mathematics of derivatives See for example Problems 81 90 in Section 15 SOME ADVICE Review the group work The problems in the group work were written by the same person who will be writing your midterm and were deemed important enough to spend class time on Try re doing the group works blank copies are available on the webpage and then go back and check the solutions Do not simply read over my solutions Focus on the areas in which you are least comfortable If there is a particular section or two of the chapter that you are not as con dent on perhaps spend a little extra time thereimaybe try to re do a few extra problems from that section Find at least one other person in the class to study with When you are talking about mathematics7 you remember it better When you ask someone else a question7 it helps that person check their own understanding and also helps you to get your questions answered So don7t be afraid to ask questions Here7s an idea Find a study buddy and write them a quiz of 3 4 derivative problems Work out the answers to your quiz rst7 and then give the quiz to your buddy Your buddy can do the same for you Then exchange answers and see how well you each did Feel free to ask me as many questions as you want I would also be happy to pop quiz7 you on a topic For example7 you can come ask me to give you some derivatives7 and you can practice right on the spot Or I can give you some limit problems Whatever you want7 actually Basically7 l7m here to help and I want everybody to do well7 so please don7t be shy Practice Practice Practice Take the sample midterm as if it was a real test no notes7 no books7 no help from friends7 no outside distractions PRACTICE Here is a list of some good problems from the book that you might like to practice with After that7 you can try the sample midterm that I posted Again7 print out the blank test and try itisolutions will be posted later7 but you should not read those until you have tried to solve the problems on your own Please do not expect the actual midterm to be the same as the sample midterm While some of the problems might be similar7 I make no promises in this regard and you should study everything well7 not just the stu that appears in the sample midterm Good Problems from the Book 757575757575 111 19 26 112 113 72 77 121 15 22 23 122 38 39 49 54 130 7a7 9a7 15a 57 vvvvvvvvvvvvvv 132 133 145 14 15 15 16 IC NCTJCTJ D 33 37 40 21ltcgt 36 use di erence quotient no short cuts 27 40 61 63 83 86 12 21 23 40 99 16 174 175 176 181 18 183 112 117 4 8 22 25 28 30 39 60 72 76 9 12 21 27 40 48 54 Math 135 Midterm III Information Spring 2009 WHEN Monday May 4 RULES No books or notes You may bring a scienti c nongraphing calculator to use for the exam It is recommended that you ensure that your calculator has suf cient battery power because if your calculator fails during the exam you will have to continue the exam without it Please bring your own calculatorllll Do not show up expecting me to supply one to you If you don7t bring one you will have to take the exam Without a calculator REVIEW SESSIONS Saturday May 2 1 3 pm MH 285 Sunday May 3 3 5 pm MH 285 EXTRA OFFICE HOURS Thursday April 30 4 6 pm and 8 10 pm Friday May 1 9 10 am and 11 1 pm Saturday May 2 11 1 pm and 7 9 pm Sunday May 3 12 3 pm Monday May 4 9 10 am and 11 1230 pm COVERAGE The midterm will cover 0 Chapter 3 Sections 3 4 and 6 0 Chapter 4 Sections 1 5 and 0 Chapter 5 Sections 1 2 and 4 STUDYING Here is an overview of the topics we have covered You should be comfortable with all of the following phrases below 0 Exponential Growth growth rate doubling time examples population con tinuously compounded interest etc o Exponential Decay half life Newton7s Law of Cooling examples murder scene Elasticity of demand using elasticity to maximize revenue Riemann sum using areas of rectangles to approximate the area under a curve The more rectangles the better the approximation Antiderivatives and Integration lntegrating a rate of change f in order to nd the total change in lnde nite lntegral and De nite lntegral know the di erencel lntegration formulas See Theorems 3 and 4 on p 406 407 lnitial Conditions to solve for the constant arising from anti differentiation Distance velocity and acceleration problems Area under a curve de nite integrals Area bounded between two curves Average value of f on the interval ab u substitution Supply and Demand curves Equilibrium point Consumer surplus Producer surplus Future value lump sum vs continuous ow Present value lump sum vs continuous ow Probability experiment sample space event Continuous random variables probability density functions Distributions uniform exponential It is not enough to simply know what all of the words and phrases above mean You need to be able to solve problems that involve the concepts that occur in the list above To help you l7ve made a list of some of the most important things that you should be able to do con dently without notes books etc by the time you reach the midterm THINGS TO BE ABLE TO DO Be comfortable using the exponential growth model Pt P061 to solve prob lems Here k gt 0 In particular you need to be able to solve this equation for an unknown such as t P0 k and so on given other information instead Be able to compute doubling time for an exponential growth model Applications include population growth continuously compounded interest etc Be comfortable using the exponential decay model Pt P061 to solve prob lems The difference here is that k lt 0 In particular you need to be able to solve this equation for an unknown such as t P0 k and so on given other in formation instead Be able to compute half life for an exponential decay model Applications include Carbon dating radioactive decay population decline etc Know the formula for Newton7s Law of Cooling and be able to solve problems with it such as the Criminology applications Problems 32 33 on p362 363 Be able to compute the elasticity of demand Be able to use the elasticity of demand to determine the maximum revenue Be able to approximate the area under a graph by using rectangles of a speci ed width or a speci ed number of them Understand that using more rectangles each of smaller width to approximate area results in a better approximation Given the rate of change f of some quantity be able to use integration to nd the total change in f on an interval 1 b eg Problems 5 12 in Section 41 Be able to perform anti differentiation in your sleep This includes the following important forms k1 s xkdz o k1 for k 31 71 1 7d lnzC s 1 e mdx 76 C a Understand the difference between an inde nite integral and a de nite integral The former gives you a family of functions7 whereas the latter gives you a number Understand the circumstances under which the area interpretation of the de nite integral breaks down ie when f goes below the z axis The value of the integral could come out negative7 whereas area is always a physical quantity that must be positive For inde nite integrals7 be able to solve for the arbitrary constant 0 that arises in the answer if additional information ie an initial condition is given Be able to start with the acceleration at and perform integrations to get the velocity 11t and the position 5t Again7 initial conditions can be used to solve for the constants that arise in the anti differentiation Be able to nd where two graphs cross by setting their equations equal to each other Be able to nd the area bounded between two graphs7 including the possibility that the graphs cross at some point on the domain of integration Be able to nd the average value of a function fx over an interval 17 Be able to solve de nite or inde nite integrals by u substitution Be able to determine the equilibrium point on a graph of supply Sx and demand Know the equations for producer surplus and consumer surplus7 and be able to compute these quantities Be able to compute future value of an investment both lump sum and contin uous ow over time under continuous compounding Equation 1 on p 476 and box above Example 2 on p 478 Be able to compute present value of an investment both lump sum and contin uous ow over time under continuous compounding Boxed at top of p 481 and top of p 482 Be able to compute the probability of a given event in an experiment Know what a continuous random variable is Know what a probability density function is Be able to verify if fx is a probability density function Be comfortable working with uniform probability distributions Be comfortable working with exponential probability distributions SOME ADVICE Practice Word Problems This didn7t go as well on the previous midterm Word problems like Half life7 Time of murder computations7 Population growth7 Problems like 5 12 in Section 417 Present and Future value of investments Integrate integrate integrate There will be lots of integration on the midtermithere is no such thing as too much practice Review the group work especially Group Works 7 and 8 The problems in the group work were written by the same person who will be writing your midterm7 and were deemed important enough to spend class time on Try re doing the group works blank copies are available on the webpage7 and then go back and check the solutions Do not simply read over my solutions Focus on the areas in which you are least comfortable If there is a particular section or two of the chapter that you are not as con dent on7 perhaps spend a little extra time thereimaybe try to re do a few extra problems from that section Find at least one other person in the class to study with When you are talking about mathematics7 you remember it better When you ask someone else a question7 it helps that person check their own understanding and also helps you to get your questions answered So don7t be afraid to ask questions Here7s an idea Find a study buddy and write them a quiz of some integrals Work out the answers to your quiz rst7 and then give the quiz to your buddy Your buddy can do the same for you Then exchange answers and see how well you each did Feel free to ask me as many questions as you want I would also be happy to pop quiz7 you on a topic For example7 you can come ask me to give you some functions to graph7 and you can practice right on the spot Or I can give you some word problems Whatever you want Basically7 l7m here to help and I want everybody to do well7 so please don7t be shy Practice Practice Practice Take the sample midterm as if it was a real test no notes7 no books7 no help from friends7 no outside distractions PRACTICE Here is a list of some good problems from the book that you might like to practice with After that7 you can try the sample midterm that I posted Again7 print out the blank test and try itisolutions will be posted later7 but you should not read those until you have tried to solve the problems on your own Please do not expect the actual midterm to be the same as the sample midterm While some of the problems might be similar7 I make no promises in this regard and you should study everything well7 not just the stu that appears in the sample midterm Good Problems from the Book Section 33 971329 Section 34 1711719733 Section 36 378715 Section 41 5925 Section 42 379152127395369777781 Section 43 5713717719731735745749759778783 Section 44 11714721729733735739747755 Section 45 5791972573974175776773 Section 51 3710 Section 52 578715 Section 54 7715721735

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